SOA Exam P Question: $P$ is a random point on the Cartesian Coordinate Plane. Find the variance of the area of a circle formed by $P$.?

Question

Caution: This problem was "passed down" to me and I think the wording was altered or lost along the way. I will post the problem as I have it and then make suggestions on what I think it should be. I am interested if anybody has seen this problem before and knows the correct wording or if anybody can "fill in the gaps" so that the problem makes sense.Problem: $P$ is a random point on the Cartesian Coordinate plane. $P$ is uniformly distributed between 0 and 3. What is the variance of the circle formed by $P$.**Edit**: What is the variance of the **area** of the circle formed by $P$.Here are my thoughts: It doesn't make sense to say that $P$ is uniformly distributed between 0 and 3 since $P=(x,y)$ is an ordered pair. So this problem would only make sense if we have that $P=(x,y)$, where $X,Y$~ Uniform(0,3), or $||P||$~Uniform(0,3), where $||P||=\\sqrt{x^2+y^2}$. In either case we get different variances.I have tried this problem under both assumptions am I am beginning to believe that it's the latter case. If so, then I would write $A=\\pi||P||^2$ and then do a random variable transformation to get the pdf of $A$. Then I would find the first and second moments of $A$ and thus have the variance.What do you think? I am interested in knowing either a correct/appropriate wording of the problem or a solution (or both).

Answer 1

Most likely it is circle with radius X. V(pi* X^2). You will need V(X^2)